Chapter IV: Freeze-cast Honeycomb Structures via Gravity-Enhanced Con-
4.5 Conclusion
of dendrites. Furthermore, convective flow leads to temperature homogenization in the liquid phase, which might make the actual temperature gradient larger than the measured temperature gradient in Figure 4.1g at the solid-liquid interface. This would also contribute to a reduction in the degree of constitutional supercooling [4]
and cells are more likely to grow.
To assess the length scale of the preceramic polymer transport by convection, the porosity of the conventional freeze-cast samples and convection-enhanced freeze- cast samples were measured. Four specimens each with a thickness of ∼1.9 mm (corresponding to∼2.5 mm in the liquid phase) were sectioned from each pyrolyzed sample (top, middle-top, middle-bottom, and bottom). In order to show porosity variations along the direction of gravity, porosity of top section was subtracted from porosity of three sections (middle-top, middle-bottom, and bottom) and these differences are shown in Figure 4.4f. The differences are approximately±1 %, and no consistent trend can be observed in either freezing direction. It is likely that the variation in porosity is due to the measurement error in the Archimedes’ method.
This implies that the distance over which the preceramic polymer is transported by convection during convection-enhanced freezing is less than 2.5 mm in the solution.
Even though transport of the solute appears to be limited to the near vicinity of the freezing front rather than throughout the entire liquid phase, constitutional supercooling is known to take place just ahead of the solid-liquid interface. Hence, even this local solute transport reduces the degree of the constitutional supercooling, resulting in morphological and size changes of dendritic pores.
References
[1] Lionel Alexander Bethune Pilkington. “Review lecture: the float glass pro- cess”. In: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences314.1516 (1969), pp. 1–25.
[2] Jennifer A. Lewis. “Colloidal processing of ceramics”. In: Journal of the American Ceramic Society83.10 (2000), pp. 2341–2359.
[3] S.M. Copley et al. “The origin of freckles in unidirectionally solidified cast- ings”. In:Metallurgical transactions1.8 (1970), pp. 2193–2204.
[4] R. Jansen and P.R. Sahm. “Solidification under microgravity”. In: Materials Science and Engineering65.1 (1984), pp. 199–212.
[5] P.J. Prescott and F.P. Incropera. “Magnetically damped convection during solidification of a binary metal alloy”. In:Journal of Heat Transfer(1993).
[6] Jose Eduardo Spinelli, Ivaldo Leao Ferreira, and Amauri Garcia. “Influence of melt convection on the columnar to equiaxed transition and microstructure of downward unsteady-state directionally solidified Sn–Pb alloys”. In: Journal of Alloys and Compounds384.1-2 (2004), pp. 217–226.
[7] José E. Spinelli et al. “Influence of melt convection on dendritic spacings of downward unsteady-state directionally solidified Al–Cu alloys”. In:Materials Science and Engineering: A383.2 (2004), pp. 271–282.
[8] Natalia Shevchenko et al. “Chimney formation in solidifying Ga-25wt pct In alloys under the influence of thermosolutal melt convection”. In:Metallurgi- cal and Materials Transactions A44.8 (2013), pp. 3797–3808.
[9] B. Drevet et al. “Solidification of aluminium–lithium alloys near the cell/dendrite transition-influence of solutal convection”. In: Journal of crystal growth 218.2-4 (2000), pp. 419–433.
[10] José Eduardo Spinelli, Otávio Fernandes Lima Rocha, and Amauri Gar- cia. “The influence of melt convection on dendritic spacing of downward unsteady-state directionally solidified Sn-Pb alloys”. In: Materials Research 9.1 (2006), pp. 51–57.
[11] Kristen L. Scotti et al. “The effect of solidification direction with respect to gravity on ice-templated TiO2microstructures”. In:Journal of the European Ceramic Society39.10 (2019), pp. 3180–3193.
[12] Kristen L. Scotti et al. “Directional solidification of aqueous TiO2suspensions under reduced gravity”. In:Acta Materialia124 (2017), pp. 608–619.
[13] Xiaomei Zeng, Noriaki Arai, and Katherine T. Faber. “Robust Cellular Shape- Memory Ceramics via Gradient-Controlled Freeze Casting”. In: Advanced Engineering Materials21.12 (2019), p. 1900398.
[14] Maninpat Naviroj. “Silicon-based porous ceramics via freeze casting of pre- ceramic polymers”. PhD thesis. Northwestern University, 2017.
[15] William W. Mullins and R.F. Sekerka. “Stability of a planar interface during solidification of a dilute binary alloy”. In: Journal of applied physics 35.2 (1964), pp. 444–451.
[16] Sylvain Deville, Eduardo Saiz, and Antoni P. Tomsia. “Ice-templated porous alumina structures”. In:Acta materialia55.6 (2007), pp. 1965–1974.
[17] W.A. Tiller et al. “The redistribution of solute atoms during the solidification of metals”. In:Acta metallurgica1.4 (1953), pp. 428–437.
C h a p t e r 5
COARSENING OF DENDRITES IN FREEZE-CAST SYSTEMS
The work was done in collaboration with Tiberiu Stan, Sophie Macfarland, Peter W. Voorhees, Nancy Senabulya, Ashwin J. Shahani, and Katherine T. Faber. N.
Arai designed the systems for study, fabricated and characterized freeze-cast ceram- ics using SEM and mercury intrusion porosimetry, and wrote the majority of the manuscript. N. Senabulya and A Shahani performed X-ray computed tomography (XCT). T. Stan and S. Macfarland analyzed XCT datasets. K. Faber and P. Voorhees supervised this work.
5.1 Introduction
In this chapter, our focus extends to the morphological evolution of the frozen crystals over time. Coarsening, also known as Ostwald ripening, is a phenomenon which occurs in two-phase systems such as alloys and metal oxides [1], and this is driven by the reduction of interfacial energy to minimize the free energy of the system.
The total interfacial area is decreased through mass transport, which is driven by the concentration gradient resulting from a large interfacial mean curvature to a small interfacial mean curvature due to the Gibbs-Thomson effect:
𝐶𝐿 =𝐶∞+𝑙𝐶𝐻 (5.1)
where
𝐻 = 1
2(𝜅1+𝜅2)
and C𝐿 is the composition of liquid at the solid-liquid interface, C∞is the composi- tion of the liquid at flat solid-liquid interface, l𝑐is the capillary length, and H is the mean curvature of interfaces. H is determined by the two principle curvatures, 𝜅1 and𝜅2. Coarsening of alloys has been extensively studied in systems ranging from simple spherical geometries [2] to complex interconnected structures such as den- drites [3, 4]. Coarsening studies span from theory [5] to modeling [6, 7] to in-situ and ex-situ experimental studies [8, 9, 10, 11]. Two important results on coarsening of dendrites are highlighted here. First, Bower et al. found that secondary dendritic arm spacing,𝜆2, increases with coarsening time as:
𝜆2∼𝑡1/3
𝑓 (5.2)
where t𝑓 is local solidification time [12]. Second, Kammer et al. reported that the dendritic structures turned into cylinders or cylindrical-like shapes after coarsening Pb-Sn alloys and Al-Cu alloys for four days and three weeks, respectively [4]. These two observations motivate this work to apply coarsening to freeze casting in order to control the morphology and size of dendritic pores.
Studies of coarsening in freeze-cast systems are limited. Pawelec et al. investigated low-temperature ice annealing in a collagen suspension, and observed coarsened microstructures after twenty hours of annealing [13]. Liu et al. examined coarsening of camphene crystals in freeze casting of bioactive glass to obtain a controllable pore diameter, ranging from 15µm to 160µm [14]. Both were restricted to pore size measurements and qualitative image analysis. Hence, there remains a gap between these observations and what is understood at a fundamental level in alloy systems.
Furthermore, these studies were conducted using suspension-based freeze casting, where suspended colloids or powders and dissolved additives such as dispersants and binders make a comparison to alloy systems challenging and complex.
This study focuses on solution-based freeze casting and investigates the evolution of dendrites during isothermal coarsening and its effects on dendritic pore mor- phology and size. By varying time and temperature, coarsening phenomena were explored using scanning electron microscopy and mercury intrusion porosimetry.
To gain further insight into the coarsening processes in freeze-cast systems in three dimensions, X-ray computed tomography enabled us to quantitatively analyze mor- phologies and directionality by Interfacial Shape Distributions (ISD) and Interfa- cial Normal Distributions (IND). By coupling images, pore size distributions with tomography-derived ISDs and dendritic pore directionality through their INDs, our studies provide new understanding into coarsening in freeze-cast systems, allow comparisons with coarsening behavior of alloy system, and offer an additional means for pore network tailorability.
5.2 Experimental methods 5.2.1 Processing
A polysiloxane (CH3-SiO1.5, Silres®MK Powder, Wacker Chemie) preceramic poly- mer was dissolved in cyclohexane (C6H12, Sigma-Aldrich), with compositions of preceramic polymer of 20 wt.% and 30 wt.%. After a homogeneous solution was
Figure 5.1: Schematic of the gradient-controlled freeze casting setup
obtained by stirring, a cross-linking agent (Geniosil®GF 91, Wacker Chemie) was added in concentrations of 1 wt.% and 0.75 wt.% in 20 and 30 wt.% solutions, respectively, and stirred for an additional 5 minutes. Subsequently, the polymer so- lution was degassed for 10 minutes to prevent air bubbles during freezing. Freezing was done using gradient-controlled freeze-casting setup as described in Chapter 3 (Figure 5.1). All samples were frozen at freezing front velocities of 15 µm/s for 20 and 30 wt.% solutions, and temperature gradients of ∼2.6 K/mm to maintain homogeneous pore structures.
To induce coarsening after freezing was completed, the top and bottom thermo- electrics were set to temperatures close to the liquidus temperature of the solution (2◦C or 4◦C for 20 wt.% solution and 3◦C for 30 wt.% solution) and held for up to 5 hrs. To determine time for frozen samples to reach the prescribed temperature, a type K thermocouple was used to measure temperature of the samples during coars- ening1. After coarsening, the samples were cooled to -30 ◦C to re-freeze. Once frozen, the samples were placed in a freeze drier where the solvent crystals were completely sublimated. After freeze drying, the polysiloxane green bodies were pyrolyzed in argon at 1100◦C for four hours with a 2◦C /min ramp rate to convert the preceramic polymer into silicon oxycarbide (SiOC). This resulted in a porosity of ∼77% for the 20 wt.% solution and 64% for 30 wt.% solution. The resulting sample dimensions were approximately 9.5mm in height and 18mm in diameter.
1Generally, type K thermocouples have an accuracy of±1.1◦C or larger [15].
5.2.2 Characterization
Pore structures were observed using scanning electron microscopy (SEM). Longitu- dinal and transverse cross-sections were prepared using a diamond saw and imaged.
Pore size distributions were measured using mercury intrusion porosimetry (MIP).
All samples for MIP were machined with a core drill (∅= 15.9 mm) to remove the edges, and a∼1.8 mm disk was sectioned from the center of the sample.
X-ray computed tomography (XCT) was performed on selected samples to quanti- tatively measure the morphological evolution of dendrites via Absorption Contrast Tomography (ACT) on a laboratory X-ray microscope (XCT; Zeiss Xradia Versa 520, Carl Zeiss AG, Oberkochen, Germany) at the Michigan Center for Materials Characterization. Three samples (h =∼5 mm, ∅=∼1.2 mm) were chosen for this analysis: a control sample without coarsening, one coarsened at 2◦C for one hour, and another coarsened at 4◦C for three hours. During the ACT measurement, each sample was positioned 5.1 mm in front of a polychromatic X-ray source tuned to 40 kV, 3 W, and 75µA. The X-ray beam interacted with a sample volume of 1025µm x 1132µm x 1090µm. A series of 1601 X-ray projection images was collected at 0.2◦ intervals while the sample rotated through 360◦at exposure times of 1.1s per projection. A scintillator downstream from the sample converted the X-ray projec- tion images into visible light images and a 4X objective lens magnified the visible light image before coupling it to the 2k x 2k CCD detector placed 23.5 mm away from the sample. With the CCD operating at a pixel binning of 2, a scan pixel size of 1.2µm/voxel was achieved. The collected projection images were reconstructed using a filtered back projection algorithm in the Scout and Scan software provided by Zeiss Xradia Inc. to create a virtual 3D volume of the sample. Worth noting is that phase retrieval [16] was not necessary because there was sufficient contrast between the SiOC matrix and the pore network in the traditional absorption-based images. The SiOC matrix is a light gray and the pore network is a dark gray (Figure 5.2).
The control sample (without coarsening) was segmented using Otsu’s method [17]
in MATLAB. Although Otsu’s method is computationally straightforward and the preferred segmentation approach, it was not successful on the 2◦C and 4◦C coars- ened datasets due to the presence of debris and bright spot artifacts at random sections throughout the reconstructions. The coarsened datasets were instead seg- mented using a convolutional neural network (CNN) machine learning approach as described by Stan et al. [18, 19]. First, 35 representative slices were selected from
Figure 5.2: Cross-section of XCT data from (a) a control sample, (b) a sample coarsened at 2◦C for one hour, and (c) a sample coarsened at 4◦C for three hours.
Scale bar: 200µm.
each reconstruction to include sections of debris and bright spots and split into three categories: 20 images for training, 10 images for validation, and 5 images for testing.
Each image was then segmented using a combination of thresholding and manual cleaning using the GIMP software. These ground truth segmentations (along with the original images) were used to train CNNs with the SegNet architecture using the PyTorch framework. Each CNN was trained for 100 epochs on the Quest supercom- puter at Northwestern University. The CNNs each achieved 99.4% segmentation accuracy when applied to test images from the 2◦C and 4◦C coarsened datasets.
MATLAB was used for all post-segmentation analysis. It was found empirically that 120µm-thick sections (100 z-slice images) of each XCT dataset were large enough to capture the defining morphological features, yet small enough to be computation- ally manageable. All three segmented datasets were meshed and smoothed using the “smoothpatch” function. The control and 2 ◦C datasets were smoothed for 5 iterations, while the coarser 4◦C dataset was smoothed for 15 iterations. Principle curvatures (𝜅1 and 𝜅2) and normal vectors were calculated at each of the triangu- lar patches. Their respective frequencies within the microstructures are plotted as interface shape distributions (ISD) and interface normal distributions (IND).
5.3 Analysis of XCT images
5.3.1 Interfacial Shape Distribution (ISD)
The quantitative analysis of morphological evolution of dendrites (or resulting pores) was carried out by measuring the curvature of the interfacial patches. First, two invariants of the curvature tensor,𝜅𝑖 𝑗, were measured. With this measurement, the mean curvature,𝐻 is established:
Figure 5.3: A map of interfacial shapes of patches for the Interfacial Shape Distri- bution (ISD). This is a modified figure from ref. [20].
𝐻 =tr{𝜅𝑖 𝑗} = 1
2(𝜅1+𝜅2) (5.3)
where the two principle curvatures, 𝜅1 and 𝜅2, the minimum and maximum prin- ciple curvatures, respectively, can be determined to construct the interfacial shape distribution (ISD). The ISD is presented as a contour plot to map the probability of finding a patch with a given pair of principal curvatures (Figure 5.3). Since𝜅2is the maximum principle curvature of the patches, the entire distribution must reside to the left of the𝜅1=𝜅2line. The plot can be divided into four regions. For dendritic porous materials:
• Region 1 represents positive𝜅1and𝜅2and the interface patches are concave toward the solid (SiOC walls).
• Regions 2 and 3 represent 𝜅1 < 0 and 𝜅2 > 0, and interface patches are saddle shaped. Region 2 embodies interface patches which are strongly curved toward the pores whereas region 3 signifies interface patches which are strongly curved toward the solid.
• Region 4 represents negative𝜅1and𝜅2and interface patches are convex toward the solid.
All the principle curvatures were normalized with respect to the specific interface area, S𝑠, which is the total surface area of the interface divided by the volume of the dendrites, or equivalently the volume of pores. This normalization is necessary for mapping probability distributions such that microstructures with different coars- ening conditions can be compared and inspected for self-similarity. One hundred slices of images, which represent 120µm of the sample in freezing direction with a diameter of roughly 1.2 mm, were used for analysis. Since the samples were frozen under constant freezing front velocity and temperature gradient and other 100 slices from different section show similar ISD, 100 slices are assumed to be sufficient to represent the whole structures.
5.3.2 Interfacial Normal Distribution (IND)
The Interfacial Normal Distribution (IND) is a contour plot which shows the proba- bility distribution of the orientation of normals to interfacial patches, and is useful in determining the directionality of dendrites, or in this study, directionality of pores.
First, the orientation of the interfacial normals to patches are determined and stored in a unit reference sphere, in which their origins sit in the center of the sphere and their ends sit in the surface of the sphere. Then, they are projected on a 2D plane, which is tangent to the sphere and, in this case, perpendicular to the direction of the freezing. The projection used in this study is an equal-area projection. Two simple cases can be considered as examples. If the porous structure has perfectly spherical shapes, the orientation of the normals is isotropic, which results in a uniform prob- ability distribution in the IND. In contrast, in the case of cylindrical pores perfectly aligned along [001] direction, the probability distribution in the IND concentrates at the outer rim of the projection. For off-axis aligned pores, an arc-like band appears across the IND.
Figure 5.4: SEM images showing (a, b) control sample, and sample coarsened at (c, d) 2◦C for one hour, (e, f) 2◦C for three hours, (g, h) 4◦C for one hour, and (i, j) 4◦C for three hours. Inset images in (a) and (b) show primary pore and secondary pores, respectively, as indicated by red arrows, (scale bar: (a) 60µm and (b) 40µm).
Transverse images and longitudinal images show cross-sections perpendicular and parallel to the freezing direction, respectively.
5.4 Results and discussion 5.4.1 Pore structure
Figure 5.4 shows a series of SEM images of dendritic pores as a function of coars- ening treatment beginning with the control sample as-cast and pyrolyzed (Figures 5.4a and b). Since cyclohexane dendrites template the pores, the pores (appearing black in SEM images) are the negatives of dendrites [21, 22]. The transverse image (perpendicular to the solidification direction) in Figure 5.4a shows primary pores templated by primary dendrites (red arrows), and secondary pores templated by secondary dendrite arms. Tertiary pores are occasionally observed in regions where primary interpore spacings are large. The four-fold symmetry of dendritic pores is consistent with the cubic structure of cyclohexane crystals [23]. The longitudinal image (approximately parallel to the solidification direction) (Figure 5.4b) shows the cutaway view of dendritic pores, where the red arrows in inset image indicate secondary pores. When the dendrites are coarsened at 2◦C for one hour, there is an increase in both primary and secondary pore sizes as shown in Figures 5.4c and d. After three hours of coarsening at 2 ◦C, the transverse image shows larger do- mains of honeycomb-like structures (Figure 5.4e) although the secondary pores are still present as noted in the longitudinal image (Figure 5.4f). When the coarsening temperature is increased to 4◦C, morphological evolution proceeds at a higher rate (Figures 5.4g-j). Coarsening for one hour yields larger domains of the honeycomb structure in the transverse direction (Figure 5.4g) while secondary pores are still noted in the longitudinal image (Figure 5.4h). After three hours of coarsening at 4◦C, the majority of secondary pores disappear in the longitudinal image (Figure 5.4j), producing a largely honeycomb-like structure. The morphological evolution of dendritic pores observed in this solution-based freeze casting agrees well with what has been reported in coarsening of dendrites in alloys [10, 4], where dendrites evolve into cylindrical morphologies. In addition to the overall morphological change from dendritic pores to cellular pores, these SEM images further reveal the morphological change of primary pores and secondary pores. The transverse image of the control sample shows four-fold symmetric primary pores (Figure 5.4a). When the structures are coarsened, primary pores evolve to circular-like shapes. See also secondary pores in the longitudinal images in Figure 5.5 comparing the control sam- ple and the sample coarsened at 4◦C for one hour as an example. While the sides of secondary pores exhibit curvature, the top and bottom faces of secondary pores are nearly flat. After coarsening, these flat surfaces disappear, and the secondary pores became circular in cross-section. Longer coarsening time (five hours) at 4